can't get a clear wiring diagram for this unit and don't find one included in likely hidey holes on the machine. or glued onto metal surface anywhere.

I did get a poor quality photo of a diagram for an lgn 1000 which is probably somewhat similar.

I found the label for the flame safety which appears to be in parallel with the gas valve coils and in series with the ignitor. so I'm guessing that the switch is NC and once the ignitor heats the flame sensor, which I assume is photo based, opens thus leaving the only path for current through the higher resistance of the coils. Although the main and helper coils make through the ignitor, only the hold coil seems to have a direct neutral so I'm wondering if the ignitor is supposed to stay heated during that transitional moments as the valve opens.

I'm assuming my present problem is one of both of the coils which seems the most common issue if the ignitor is actually working and the ignitor works and then does go out which suggests the safety is opening. Will I likely find the coil open or do they fail because of loss of resistance?

upon thinking about the whole thing some more i realize the ignitor is probably doubling as a continuing flame sensor. that is why the coils are in series with it so the resistance must be low when hot and increase when it goes cold and that would diminish voltage across the holding coils to a point where they close.

i agree, no need to overthink, i just like the thinking work.

also, on my coils one read open so that was pretty obvious. the set set me back 15 bucks and ya only got to get as far as whirlpool dryer coils and they are coming off the shelf.

The gas safety valve operation is hard to explain. You're right, the flame sensor is closed and opens from the heat of the ignitor. When the flame sensor opens the voltage is sent to the coils. The valve opens and the flame ignites. The heat from the flame continues to keep the flame sensor open.